A note on edge irregularity strength of Dandelion graph

Abstract

For a simple graph $G$, a vertex labeling $\varphi :V(G)\to \{1,2,\dots ,k\}$ is called $k$-labeling. The weight of an edge $xy$ in $G$, written $w_{\varphi }(xy)$, is the sum of the labels of end vertices $x$ and $y$, i.e., $w_{\varphi }(xy)=\varphi (x)+\varphi (y)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two different edges $e$ and $f$, $w_{\varphi }(e)\ne w_{\varphi }(f)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, written $es(G)$. In this note, we find the exact value of edge irregularity strength of Dandelion graph when $\Delta (G)\ge \lceil \frac{|E(G)|+1}{2}\rceil$; and determine the bounds when $\Delta (G)<\lceil \frac{|E(G)|+1}{2}\rceil$.

To appear in the Southeast Asian Bulletin of Mathematics